1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a prototypical natural unbounded operator...

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by ...

In this paper, in an arbitrary Hilbert space nonlocal boundary value problem for the Schrödinger equation with time dependent self-adjoint operator is studied. Stability estimates solution of established. To find approximate first order accuracy Rothe difference scheme and second Crank-Nicholson are constructed. these schemes have been obtained. obtain stability estimates, theory spectral repre...